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Theory and Decision

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01.03.2014

On the significance of the prior of a correct decision in committees

verfasst von: Ruth Ben-Yashar, Shmuel Nitzan

Erschienen in: Theory and Decision | Ausgabe 3/2014

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Abstract

The current note clarifies why, in committees, the prior probability of a correct collective choice might be of particular significance and possibly should sometimes even be the sole appropriate basis for making the collective decision. In particular, we present sufficient conditions for the superiority of a rule based solely on the prior relative to the simple majority rule, even when the decisional skills of the committee members are assumed to be homogeneous.

Vorheriger Artikel If nudge cannot be applied: a litmus test of the readers’ stance on paternalism

Nächster Artikel A geometric approach to revealed preference via Hamiltonian cycles

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The general uncertain dichotomous choice model can be used to study optimal informative voting in any voting body, Baharad et al. (2011, 2012); Ben-Yashar and Danziger (2011); Ben-Yashar and Kraus (2002); Ben-Yashar and Nitzan (1997); Nitzan (2010); Nitzan and Paroush (1982, 1985); Nurmi (2002); Shapley and Grofman (1984); Young (1995). Here both decisional skills and the prior of a correct collective decision are explicitly taken into account and, of course, the optimal rule need not be the SMR.

The classical social choice problems due to the existence of heterogeneous preferences, e.g., the problem of majority tyranny, Baharad and Nitzan (2002) or the difficulty of attaining a reasonable social compromise, Young (1988, 1995), can be disregarded in our setting.

CJT can be generalized to the case of heterogeneous voters. See, for example, Ben-Yashar and Zahavi (2011); Berend and Paroush (1998); Berend and Sapir (2005).

Proof: As noted above,

$$\begin{aligned} f^{*}=sign\left( \sum _{i=1}^n w x_i +\gamma \right) \end{aligned}$$

The SMR is defined as follows:

$$\begin{aligned} f^{SMR}=sign\left( \sum _{i=1}^n x_i\right) \end{aligned}$$

Therefore, the SMR is optimal if

$$\begin{aligned} \forall x\left( {\sum _{i=1}^n w x_i +\gamma >0\Leftrightarrow \sum _{i=1}^n {x_i } >0} \right) \end{aligned}$$

Notice that for $x$ such that the number of supporters in alternative 1 is $\frac{n-1}{2}$,

$$\begin{aligned}&f^{*}=sign\left( (\frac{n-1}{2}-\frac{n+1}{2})\ln \frac{p}{1-p})+\ln \frac{\alpha }{1-\alpha }\right) =sign\left( -\ln \frac{p}{1-p}+\ln \frac{\alpha }{1-\alpha }\right) =1, \\&\qquad \quad \text{ since, } \text{ by } \text{ assumption }, \alpha >p.\\&f^{\mathrm{SMR}}=sign\left( \frac{n-1}{2}-\frac{n+1}{2}\right) =-1. \text{ That } \text{ is }, f^{*}\ne f^{\mathrm{SMR}}. \end{aligned}$$

$\square $

Proof: Note that when $\alpha >1/2$ , $f^{\mathrm{PBR}}=1.$

$$\begin{aligned}&f^{*}=f^{\mathrm{PBR}}=1, if \forall x, \end{aligned}$$

$\mathop {\sum }\nolimits _{i=1}^n w x_i +\gamma >0 \quad $, and this is true for every $x$, in particular, even for $x=(-1,-1,-1,\ldots ,-1).$ This means that $\gamma -nw>0\Leftrightarrow \ln \frac{\alpha }{1-\alpha }>n\ln \frac{p}{1-p}\Leftrightarrow \frac{\alpha }{1-\alpha }>(\frac{p}{1-p})^{n}$. $\square $

For larger committees, more individuals are required to vote non-informatively to increase the probability of making the correct collective decision. In the extreme case where the PBR is the optimal rule, non-informative voting by the majority of the committee members can lead to the attainment of the maximal performance (the highest possible probability of making the correct collective choice). Clearly, strategic voting by an individual committee member is less effective than strategic voting by a subgroup of the committee members.

This advantage is decreasing with $k$ because

$$\begin{aligned}&\left( {{\begin{array}{l} {2(k+1)} \\ {k+1} \\ \end{array} }} \right) p^{k+1}\left( {1-p} \right) ^{k+1}(\alpha -p)-\left( {{\begin{array}{l} {2k} \\ k \\ \end{array} }} \right) p^{k}\left( {1-p} \right) ^{k}(\alpha -p)<0\Leftrightarrow \\&\begin{array}{l} \\ \left( {{\begin{array}{l} {2k} \\ k \\ \end{array} }} \right) p^{k}\left( {1-p} \right) ^{k}(\alpha -p)(\frac{(2k+1)(2k+2)}{(k+1)(k+1)} p\left( {1-p} \right) -1)<0\Leftrightarrow \\ \left( {{\begin{array}{l} {2k} \\ k \\ \end{array} }} \right) p^{k}\left( {1-p} \right) ^{k}(\alpha -p)(\frac{(2k+1)2p\left( {1-p} \right) -(k+1)}{(k+1)} <0\Leftrightarrow \\ 2p\left( {1-p} \right) < \frac{k+1}{2k+1}. \\ \\ \end{array}\end{aligned}$$

Austen-Smith, D., & Banks, J. S. (1996). Information aggregation, rationality and the Condorcet Jury Theorem. American Political Science Review, 90(1), 34–45.CrossRef

Baharad, E., Goldberger, J., Koppel, M., & Nitzan, S. (2011). Distilling the wisdom of crowds: Weighted aggregation of decisions on multiple issues. Journal of Autonomous Agents and Multi-Agent Systems, 22(1), 31–42.CrossRef

Baharad, E., Goldberger, J., Koppel, M., & Nitzan, S. (2012). Beyond Condorcet: Optimal judgment aggregation using voting records. Theory and Decision, 72(1), 113–130.CrossRef

Baharad, E., & Nitzan, S. (2002). Ameliorating majority decisiveness through expression of preference intensity. American Political Science Review, 96(4), 745–754.CrossRef

Ben-Yashar, R., & Danziger, L. (2011). On the optimal allocation of committee members. Journal of Mathematical Economics, 47, 440–447.CrossRef

Ben-Yashar, R., & Kraus, S. (2002). Optimal collective dichotomous choice under quota constraints. Economic Theory, 19, 839–852.CrossRef

Ben-Yashar, R., & Milchtaich, I. (2007). First and second best voting rules in committees. Social Choice and Welfare, 29(3), 453–480.CrossRef

Ben-Yashar, R., & Nitzan, S. (1997). The optimal decision rule for fixed-size committees in dichotomous choice situations: The general result. International Economic Review, 38(1), 175–186.CrossRef

Ben-Yashar, R., & Zahavi, M. (2011). The Condorcet Jury Theorem and extension of the Franchise with rationally ignorant voters. Public Choice, 148, 435–443.CrossRef

Berend, D., & Paroush, J. (1998). When is Condorcet’s Jury Theorem valid. Social Choice and Welfare, 15, 481–488.CrossRef

Berend, D., & Sapir, L. (2005). Monotonicity in Condorcet Jury Theorem. Social Choice and Welfare, 24, 83–92.CrossRef

de Condorcet, N. C. (1785). Essai sur l’application de l’analyse a la probabilite des decisions rendues a la pluralite des voix. Paris, 20, 27–32.

Feddersen, T., & Pesendorfer, W. (1998). Convicting the innocent: The inferiority of unanimous Jury Verdicts under strategic voting. American Political Science Review, 92, 23–35.CrossRef

Koppel, M., Argamon, S., & Shimon, A. L. (2012). Automatically categorizing written texts by author gender. Ramat Gan: Department of Computer Science, Bar Ilan University.

McLennan, A. (1998). Consequences of the Condorcet Jury Theorem for beneficial information aggregation by rational agents. American Political Science Review, 92, 413–418.CrossRef

Miller, N. (1996). Information, individual errors, and collective performance: Empirical evidence on the Condorcet Jury Theorem. Group Decision and Negotiation, 5, 211–228.CrossRef

Nitzan, S. (2010). Collective preference and choice. Cambridge: Cambridge University Press.

Nitzan, S., & Paroush, J. (1982). Optimal decision rules in uncertain dichotomous choice situations. International Economic Review, 23(2), 289–297.CrossRef

Nitzan, S., & Paroush, J. (1985). Collective decision making: An economic outlook. Cambridge: Cambridge University Press.

Nurmi, H. (2002). Voting procedures under uncertainty. Berlin, Heidelberg: Springer.CrossRef

Shapley, L., & Grofman, B. (1984). Optimizing group judgmental accuracy in the presence of interdependencies. Public Choice, 43, 329–343.CrossRef

Young, P. (1988). Condorcet theory of voting. American Political Science Review, 82(4), 1231–1244.CrossRef

Young, P. (1995). Optimal voting rules. Journal of Economic Perspectives, 9(1), 51–64.CrossRef

Titel: On the significance of the prior of a correct decision in committees
verfasst von: Ruth Ben-Yashar
Shmuel Nitzan
Publikationsdatum: 01.03.2014
Verlag: Springer US
Erschienen in: Theory and Decision / Ausgabe 3/2014
Print ISSN: 0040-5833
Elektronische ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-013-9362-7

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